## Underground Maps, And Making Your Own

By way of the UK Ed Chat web site I’ve found something that’s maybe useful in mathematical education and yet just irresistible to my sort of mind. It’s the Metro Map Creator, a tool for creating maps in the style and format of those topographical, circuit-diagram-style maps made famous by the London Underground and many subway or rail systems with complex networks to describe.

UK Ed Chat is of the opinion it can be used to organize concepts by how they lead to one another and how they connect to other concepts. I can’t dispute that, but what tickles me is that it’s just so beautiful to create maps like this. There’s a reason I need to ration the time I spend playing Sid Meier’s Railroads.

It also brings to mind that in the early 90s I attended a talk by someone who was in the business of programming automated mapmaking software. If you accept that it’s simple to draw a map, as in a set of lines describing a location, at whatever scale and over whatever range is necessary, there’s still an important chore that’s less obviously simple: how do you label it? Labels for the things in the map have to be close to — but not directly on — the thing being labelled, but they also have to be positioned so as not to overlap other labels that have to be there, and also to not overlap other important features, although they might be allowed to run over something unimportant. Add some other constraints, such as the label being allowed to rotate a modest bit but not too much (we can’t really have a city name be upside-down), and working out a rule by which to set a label’s location becomes a neat puzzle.

As I remember from that far back, the solution (used then) was to model each label’s position as if it were an object on the end of a spring which had some resting length that wasn’t zero — so that the label naturally moves away from the exact position of the thing being labelled — but with a high amount of friction — as if the labels were being dragged through jelly — with some repulsive force given off by the things labels must not overlap, and simulate shaking the entire map until it pretty well settles down. (In retrospect I suspect the lecturer was trying to talk about Monte Carlo simulations without getting into too much detail about that, when the simulated physics of the labels were the point of the lecture.)

This always struck me as a neat solution as it introduced a bit of physics into a problem which hasn’t got any on the assumption that a stable solution to the imposed physical problem will turn out to be visually appealing. It feels like an almost comic reversal of the normal way that mathematics and physics interact.

Pardon me, now, though, as I have to go design many, many imaginary subway systems.